The council is this week due to debate the proposals, which would see councillors consult on how the savings should be achieved, which could mean charging or reducing the range of instruments.
Currently, instrument tuition is free in schools in Edinburgh but many other councils charge for the service.
Many local authorities in Scotland have in recent months considered cutting musical tution provision in schools. In November, Dumfries and Galloway Council unveiled plans to axe musical instrument tuition for pupils – less than a year after Midlothian Council was forced to drop similar proposals after a public outcry.
Caroline Sewell, regional officer for Scotland for the Musicians Union, who launched the petition, said: “It’s not right and it’s not fair. Cutting Edinburgh’s instrumental music service means taking opportunity away from Edinburgh’s children and young people.
“Over 40 per cent of those from low-income families say music lessons are beyond their household budgets.
“Musicians’ Union research shows that families with a total household income of less than £28,000 are half as likely to have a child learning an instrument as more affluent peers with a family income of £48,000 or more, despite similar levels of interest from both groups of children.”
She added: “Music should be available and attainable for all, whether they are the next Lewis Capaldi, Nicola Benedetti or Primal Scream, or they just want to try something new.”
In addition to cutting music tuition, the council is proposing to withdraw qualified teachers from nursery schools to save £900,000; slash budgets handed to school heads by £1.8 million; and cut the number of qualified improvement officers who give specialist advice and support to heads, saving £240,000.
Alastair Orr, who teaches brass instruments in schools in Stirling, grew up in Edinburgh, attending Portobello High School.
He said: “My parents would not have been able to afford to pay for lessons of an instrument and without free tuition I would not be a musicians today. I feel strongly that the council’s current enlightened policy of free tuition to all must continue.”
Two years ago, education secretary John Swinney said: “I am concerned by decisions by a number of them to reduce access to instrumental music tuition.”
Edinburgh Council said: “We fully recognise the benefits for young people of playing a musical instrument. As part of our consultation later in the year we will make sure that pupils from most disadvantaged backgrounds and those studying SQA music will continue to receive tuition.
“The consultation is key to shaping our future plans for the service while recognising the need to make savings.”